Parabola Equations
 

A projectile is launched at x=0 .y=h with speed V and angle θ. What is the equation of the parabola (no air drag)?

y = ax² + bx + c

a=(g/2)/(Vcosθ)² ... (g is negative)

b=tanθ

c=h


Given 2 points (x₁,y₁) and (x₂,y₂), find the equation of the parabola that passes through the origin and those 2 points

y = ax² + bx + c

a=(x₂y₁-x₁y₂)/(x₁²x₂-x₁x₂²)

b=-(x₂²y₁-x₁²y₂)/(x₁²x₂-x₁x₂²)

c=0


Given 2 points (x₁,y₁) and (x₂,y₂), find the equation of the parabola that passes through (0,h) and those 2 points

y = ax² + bx + c

a=(x₁(h-y₂)+x₂y₁-hx₂)/(x₁²x₂-x₁x₂²)

b=-(x₁²(h-y₂)+x₂²y₁-hx₂²)/(x₁²x₂-x₁x₂²)

c=h


Given 3 points (x₁,y₁), (x₂,y₂), and (x₃,y₃) find the equation of the parabola that passes through those 3 points

It gets messy.


Given parabola y = ax² + bx + c , where a<0, find the value of x and y at the apex

x = -b/(2a) ... y=c-b²/(4a)


Given parabola y = ax² + bx + c , where a<0, find the value of x and y for which the slope is -1

x = -(1+b)/(2a) ... y = c+(b+1)²/(4a)-b(b+1)/(2a)


someone please check my math

Re: Parabola Equations
 

I'm not sure if that's a challenge, or a dare.

Re: Parabola Equations
 

Reps if you find an error.

Re: Parabola Equations
 

Given the coordinates (x_p,y_p) of the apex of a parabola whose width is W at a distance D below the apex, find the equation of the parabola.

Re: Parabola Equations
 

Given 2 points (x1,y1) and (x2,y2), and the slope m1 at (x1,y1), find the equation of the parabola.

Re: Parabola Equations
 

have to ask: did you use LaTeX for those pretty pictures?

Re: Parabola Equations
 

I use Maxima to work out the math. Then I capture a PNG screenshot of the area of interest.

For post#1, all the equations were created using the editing available in vBulletin.

Re: Parabola Equations
 

Given the coordinates (0,h) of the launch point and (xp,yp) of the apex, find the equation of the parabola.

Re: Parabola Equations
 

Given the values of a, b, and c in the equation y=ax²+bx+c of a parabola, find the launch speed V and launch angle theta if the launch point is at (0,h).

Re: Parabola Equations
 

I give Ether my highest complements. It takes a certain kind of problem for me to voluntarily work on it for about 4 hours. I worked through the 1st, 5fth, and 6th sub-problems relatively quickly, but the middle three are what are stalling me. I have attempted to solve sub-problems 3 and 4 using both systems of equations and using matricies with Cramer's rule. My system of equations was having sign issues (more than can be expected considering I am comparing my solution to Ether's) and my matricies were not working from my rustiness in this area. I will continue to work tonight and post if anything changes.

Once again, props to you, Ether.

Re: Parabola Equations
 

I just finished up the solutions problems in the original post, using three pages of paper in the process. Ether, your solutions check out fine to me (albeit I have very well could have missed something). I also see what you mean by the solution being messy. I think the derivation was messier. To anyone wondering, I solved it each time with a set of matrices and Cramer's Rule. Cramer's rule is both really cool and useful. I advise you all to go check it out. I may try my hand at proving the others tomorrow. Ether, as I stated previously, this is a very good challenge. Thank you.

Re: Parabola Equations
 

Just finished up all the problems and everything checks out!

I solved most of them using row reduction then the rest were solved using MATLAB. For anyone that's currently taking a class or looking for a refresher in matrices or 2-d kinematics, try these problems out!